Ross M. Richardson's Research

It is six in the morning. The house is asleep. Nice music is playing. I prove and conjecture.
-Paul Erdos

I'm a third year graduate student at UCSD. My primary research falls under the heading of combinatorics, including graph theory and any interesting discrete structures I come across. I'm especially interested in discrete and computational geometry. Additionally, I'm still a bit enamored with geometric topology, as well as my original love of theoretical computer science.

I currently work with Dr. Fan Chung Graham and Dr. Van Vu, doing combinatorial things. Dr. Graham maintains a list of advisees/mentees, who together with Lei Wu constitute the analytic combinatorics students at UCSD math (as opposed to the algebraic/enumerative combinatorics folk). Yes, there is enough combinatorics here that we draw such distinctions. Speaking of which, UCSD is number two in combinatorics, according to US News and Conservative reports.

Here are some resources of our group.

• The Combinatorics of Large Sparse Graphs. This was a course given by Dr. Graham, and I maintain the lecture notes here for perpetuity. Note that there will be a forthcoming monograph on this subject, which should subsume these notes.
• 2004-2005 Reading Seminar List.
• 2005 Summer Seminar

So what's new? It's official, I'm staying in San Diego. Now, on to graduating. Meanwhile, more random polytopes and applications, though mostly working on really cracking the sigma-function stuff.

Additionally, 我这个夏天会去天津。在南开大学会学数学一个月。Or, for those that don't read pidgeon mandarin, I'm going to China! My generous advisor is sending me to the Center for Combinatorics at Nankai University for part of the summer. Will I survive? Tune in to find out.

I should mention the following people, who have supported me in research over my undergraduate tenure: Weiqing Gu, Sampath Kannan, Sanjeev Khanna, Elizabeth `Z' Sweedyk.

Computations and Pictures

• IMDB Graphs-Drawings and Degree Sequences
• The Sigma function of a graph.
• BGP Webgraph - Drawings and Spectra
• Fan's Graph Gallery

Papers

• PhD Thesis--Ha! (Should obtain one....)
• Graph Drawing with the Sigma Function. In progress.
• Random Circumscribing Polytopes. In progress.
• Weighted Laplacians and the Sigma Function of a Graph.Gregory Berkolaiko, Robert Carlson, Stephen A. Fulling, and Peter Kuchment (Editors), Quantum Graphs and Their Applications, Contemporary Mathematics, v. 415, AMS, Providence, RI 2006.. (with Fan Chung). PDF.
• An inscribing model for random polytopes.Submitted. (with Van Vu and Lei Wu). PDF.
• Random Inscribing Polytopes.To appear, European Journal of Combinatorics, Special Ed. 2005. (with Van Vu and Lei Wu). PDF.
• Senior Thesis-Designing Rigid Motions:Computing Curves in Lie Groups. Advised by Weiqing Gu, Harvey Mudd College

Presentations

• Graphs and Probability; A Perfect Matching. Food for Thought Graduate Seminar, UCSD. Spring '06
• Random Inscribing Polytopes. Workshop on Probabilistic Combinatorics. BIRS, Banff, Canada. Fall '05.
• Weighted Laplacians and Sigma-Functions. Graduate Student Research Seminar. Fall '05.
• Random Polytopes, 3 lectures, Reading Seminar in Combinatorics at UCSD, Fall '04, Spring '05. Directed by Van Vu.

Conferences Attended

• BIRS workshop on Probabilistic Combinatorics, Fall 05
• AMS Sectional, UCSB
• Mt. Baldy Conference on Geometry, Algebra, and Phylogenetic Trees, Fall 04
• MSRI Conference on Discrete and Computational Geometry, Fall 03
• Mount Baldy Conference on Applied Algebra and Combinatorics, Fall 02
• Mount Baldy Conference on Differential Geometry, Fall 01

Research Links

• UCSD Geometry Seminar.
• UCSD STAR (CS Theory) Seminar
• Seminars and Colloquia at UCSD.
• MSRI Program in Discrete and Computational Geometry. This is ongoing during the Fall of '03. A great place to find the new and open in the field.
• The Open Problem Page - Problems in Computational Geometry
• Geometry Junkyard - Open Problems
• Open Problems - Graph Theory and Combinatorics
• Open Problems in Combinatorics
• Open Problems of Paul Erdos
• Lectures on Discrete Geometry. A fantastic reference for all things discrete geometric. Also a model of exposition.
• The Probabilistic Method. The are no reasonable alternatives to learning this fantastic material. Luckily, this is a good book.
• Spectral Graph Theory. Spectral Goodness. Chances are if you see me I'm carrying around some subset of the last three books.
• An introduction to Convex Geometry - K. Ball. This is a great introduction to the tools and methods of modern convex geometry, clearly written, without the need of great technical background. Highly recommended (complements Matousek's book nicely). Also, it's free!
• Volume Estimates and Rapid Mixing - B. Bollobás Geometry, Combinatorics, and Probability. Gotta love it.
• Convex Geometric Analysis - MSRI Book The modern direction of convex geometry. Lots of big names and well written documents here.
• Kim's Probabilistic Method Notes. An alternate source of prob. method which isn't derived from Alon and Spencer.

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